# Solutions

```Math 1271 Dis. 061
Quiz 8
November 15, 2005
Solutions
No Work = No Credit. Write Legibly. Box your final result.
1. 10 points If 1200 cm2 of material is available to make a box with a square base and an
open top, find the largest possible volume of the box.
Since the base of the box is a square, we let x be the length of any side of the square. Thus,
the area of the base is given by x2 . Let y be the height of the box (do not assume that the
box is a cube). The box is open (no top) thus, total surface area of the box (4 sides and the
base) is given by
A = x2 + 4xy.
From the problem statement, we know that A = 1200 cm2 thus, we have the following
constraint relationship
1200 = x2 + 4xy.
(1)
The volume of the box is given by
V = x2 y.
Solving (1) for y and substituting it into (2) yields
1200 − x2
2
.
V =x
4x
(2)
(3)
To find the maximum volume, we take a derivative of (3) with respect to x and set the result
equal to zero, to obtain
3x2
300 −
= 0.
(4)
4
Solving (4) for x, yields x = 20 cm. To find the y value, we substitute x = 20 cm into (1),
to obtain y = 10 cm. Finally, substituting x = 20 cm and y = 10 cm into (2) yields
V = 4000 cm3 .
```